Group theoretical independence of $\ell$-adic Galois representations

Abstract: Let $K/\mathbb{Q}$ be a finitely generated field of characteristic zero and $X/K$ a smooth projective variety. Fix $q\in\mathbb{N}$. For every prime number $\ell$ let $\rho_\ell$ be the representation of $\mathrm{Gal}(K)$ on the \'etale cohomology group $H^q(X_{\overline{K}}, \mathbb{Q}\ell)$. For a field $k$ we denote by $k{\mathrm{ab}}$ its maximal abelian Galois extension. We prove that there exist finite Galois extensions $k/\mathbb{Q}$ and $F/K$ such that the restricted family of representations $(\rho_\ell|\mathrm{Gal}(k_{\mathrm{ab}} F))\ell$ is group theoretically independent in the sense that $\rho{\ell_1}(\mathrm{Gal}(k_{\mathrm{ab}} F))$ and $\rho_{\ell_2}(\mathrm{Gal}(k_{\mathrm{ab}} F))$ do not have a common finite simple quotient group for all prime numbers $\ell_1\neq \ell_2$.